I like slaven41's answer a lot, but for what it's worth here are my thoughts on the subject.
In trying to understand the universe, we use analogy more often than reason. We call stuff "matter" if it is like a rock, and we call stuff "light" if it is like sunlight. The ancient Greeks thought about this a lot. Democritus thought matter was made up of indivisible "atoms", and was mostly right, except that atoms are themselves made up of smaller particles. Lucretius thought that "light" was a skin around every object, which the object kept shedding like a snake sheds its skin. He was almost entirely wrong on that subject, though he was amazingly right about the motion of falling objects, correcting Aristotle. It wasn't until modern times that people understood that light was electromagnetic in nature, with the electric and magnetic fields playing leapfrog. Because light could be polarized, the fields move up and down like...well like the waves in the ocean. Thus "light waves", which doesn't really say anything except light changes and the change has a direction. (Unlike sound waves, which compress and expand.) But light, like matter, has an indivisible kernel, a photon, so it is kind of like a particle, too.
Thus when physicists talk about the duel nature of light, they are saying it is kind of like a rock and kind of like the sea, but not really like either one. In fact, it is itself, and there is nothing else like it.
Then Einstein discovered that light can be changed to matter and matter to light, according to e = mc^2. So there is really just one "stuff" in the universe. If it travels really, really fast (at the speed of light) we call it light. If not so fast, we call it matter.
Now, I have a question. Electrons are matter, not light. But they have a wave-length. What's waving?